Let \(G=KAN\) be an Iwasawa decomposition of a real semisimple Lie group \(G\) and \(L: G\to{\mathfrak a}={\mathcal L}(A)\) determined by \(g\in K \exp L(g)N\). Then Kostant's Convexity Theorem says that \(L(aK)=\text{conv}(W \log a)\) for \(a\in A\), where \(W\) is the Weyl group associated to \({\mathfrak g}\), \({\mathfrak a}\) and conv denotes the closed convex hull. Let now \(\tau\) be an involution on \(G\) which commutes with the Cartan involution \(\theta\) determined by \(K\). Denote by \(\mathfrak q\) and \(\mathfrak p\) the \((-1)\)- eigenspaces for \(\tau\) and \(\theta\), respectively. Assume that the centralizer \(Z_{\mathfrak q}({\mathfrak c})\) of the center \(\mathfrak c\) in \({\mathfrak p}\cap{\mathfrak q}\) equals \({\mathfrak p}\cap{\mathfrak q}\). The restricted roots not vanishing on \(\mathfrak c\) are called non-compact roots and one defines a convex cone \(C_{\max}\) in \(\mathfrak a\) via the condition that all non- compact roots take non-negative values on \(C_{\max}\). If \(H\) is the connected component of the fixed point group of \(\tau\) one has an Iwasawa-type map \(L_ \tau: HAN\to{\mathfrak a}\) which coincides with \(L\) if \(\tau=\theta\). The authors main result is the following generalization of Kostant's theorem: For \(\mathbf{1}\neq a\in \exp C_{\max}\) we have \(L_ \tau(aH)=\text{conv}(W \log a)+C^*_{\max}\).